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Concept
Baker's theorem
Summary
In transcendental number theory, a mathematical discipline, Baker's theorem gives a lower bound for the absolute value of linear combinations of logarithms of algebraic numbers. The result, proved by , subsumed many earlier results in transcendental number theory and solved a problem posed by Alexander Gelfond nearly fifteen years earlier.
Baker used this to prove the transcendence of many numbers, to derive effective bounds for the solutions of some Diophantine equations, and to solve the class number problem of finding all imaginary quadratic fields with class number 1.
To simplify notation, let be the set of logarithms to the base e of nonzero algebraic numbers, that is
where denotes the set of complex numbers and denotes the algebraic numbers (the algebraic completion of the rational numbers ). Using this notation, several results in transcendental number theory become much easier to state. For example the Hermite–Lindemann theorem becomes the statement that any nonzero element of is transcendental.
In 1934, Alexander Gelfond and Theodor Schneider independently proved the Gelfond–Schneider theorem. This result is usually stated as: if is algebraic and not equal to 0 or 1, and if is algebraic and irrational, then is transcendental. The exponential function is multi-valued for complex exponents, and this applies to all of its values, which in most cases constitute infinitely many numbers. Equivalently, though, it says that if are linearly independent over the rational numbers, then they are linearly independent over the algebraic numbers. So if and is not zero, then the quotient is either a rational number or transcendental. It cannot be an algebraic irrational number like .
Although proving this result of "rational linear independence implies algebraic linear independence" for two elements of was sufficient for his and Schneider's result, Gelfond felt that it was crucial to extend this result to arbitrarily many elements of Indeed, from :
This problem was solved fourteen years later by Alan Baker and has since had numerous applications not only to transcendence theory but in algebraic number theory and the study of Diophantine equations as well.
Official source
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